Question about polynomial

Question

If the polynomial $x^3+3px+q$ has a factor of the form $(x-a)^2$, then prove that $q^2+4p^3=0$.

Answer 1

$$(x-a)^2(x-b)=x^3-bx^2-2ax^2+a^2x+2abx-ba^2$$ and by identification

$$b=-2a.$$

Then

$$q^2+4p^3=(2a^3)^2+4(-a^2)^3=0.$$

---

You can also perform the long division of the given polynomial by $(x-a)^2$ and cancel the remainder.

Answer 2

$a$ is a double root so that

$$a^3+3pa+q=3a^2+3p=0.$$

Eliminate $a$.